Naval Research LaboratoryWashington, DC 20375-5320

NRL/MR/5770--12-9382

Heave and Pitch Dynamics ofShallow Draft Surface Vehicles RogeR S. CoRteSi

Electronic Warfare Modeling and Simulation BranchTactical Electronic Warfare Division

owen thoRp geoRge pipeR

Systems Engineering Department United States Naval AcademyAnnapolis, Maryland

April 10, 2012

Approved for public release; distribution is unlimited.

i

REPORT DOCUMENTATION PAGEForm Approved

OMB No. 0704-0188

3. DATES COVERED (From - To)

Standard Form 298 (Rev. 8-98)Prescribed by ANSI Std. Z39.18

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

2. REPORT TYPE1. REPORT DATE (DD-MM-YYYY)

4. TITLE AND SUBTITLE

6. AUTHOR(S)

8. PERFORMING ORGANIZATION REPORT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

10. SPONSOR / MONITOR’S ACRONYM(S)9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES)

11. SPONSOR / MONITOR’S REPORT NUMBER(S)

12. DISTRIBUTION / AVAILABILITY STATEMENT

13. SUPPLEMENTARY NOTES

14. ABSTRACT

15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF:

a. REPORT

19a. NAME OF RESPONSIBLE PERSON

19b. TELEPHONE NUMBER (include areacode)

b. ABSTRACT c. THIS PAGE

18. NUMBEROF PAGES

17. LIMITATIONOF ABSTRACT

Heave and Pitch Dynamics of Shallow Draft Surface Vehicles

Roger S. Cortesi, Owen Thorp,* and George Piper*

Naval Research Laboratory4555 Overlook Avenue, SWWashington, DC 20375-5320

NRL/MR/5770--12-9382

Approved for public release; distribution is unlimited.

Unclassified Unclassified UnclassifiedUU 24

Roger Cortesi

(202) 767-6814

HeavePitch

This project’s objective is to develop a model of the heave and pitch dynamics for a shallow draft surface vehicle. The model developed uses a novel approach of representing the ship dynamics in the frequency domain. The model is based on theoretical first-principles involving highly nonlinear equations that depend on the water wave frequency. Compared to time domain modeling and simulation approaches, the frequency domain model provides greater insight to the surface vehicle’s behavior over a wider operating range. To validate the model, experimental data from towing tank experiments with a 3 meter, shallow draft boat is compared to theoretical model results. The experimental data compares well with the model.

10-04-2012 Memorandum

Naval Research Laboratory4555 Overlook Avenue, SWWashington, DC 20375-5320

NRL

*Systems Engineering Department, United States Naval Academy, Annapolis, MD 21402-5002

DynamicsModeling

Surface vesselWave-induced motion

57-6014-A95

ii

iii

Table of Contents

Nomenclature ............................................................................................................................................... v

1.0 Introduction ........................................................................................................................................... 1

2.0 Review of Wave-Induced Planar Motion Analysis ................................................................................. 1

3.0 Systems Analysis: Frequency Domain Model ........................................................................................ 4

4.0 Experiment ........................................................................................................................................... 10

5.0 Results ................................................................................................................................................... 13

6.0 Conclusions .......................................................................................................................................... 17

Acknowledgements ..................................................................................................................................... 17

References .................................................................................................................................................. 18

iv

v

Nomenclature Free surface displacement of wave

a Wave amplitude

w Wave frequency

e Frequency of encountered wave

K Wave number

V Vehicle velocity

Relative heading of vehicle to wave train

z Heave displacement

az Heave amplitude

z Heave phase angle relative to wave

ssz Heave steady state displacement

Pitch displacement

a Pitch amplitude

ss Pitch steady state displacement

Pitch phase angle relative to wave

m Vehicle mass

za Added mass (the mass of water excited by the vehicle’s motion)

b Heave damping coefficient

c Heave stiffness coefficient

d Pitch inertia coupling term

e Pitch damping coupling term

h Pitch stiffness coupling term

yyI Pitch axis moment of inertia

yyA Added mass of inertia

B Pitch damping coefficient

C Pitch stiffness coefficient

D Heave inertia coupling term

E Heave damping coupling term

H Heave stiffness coupling term

g Acceleration of gravity

Density of water

ak Added mass coefficient

zR Ratio of radiated wave amplitude to heave motion amplitude

nB Vehicle’s sectional beam width

vi

refz Sectional reference draft of the wetted surface

na Sectional added mass

nb Sectional damping coefficient

nc Sectional stiffness coefficient

F Wave induced force

0F Amplitude of force

F Phase angle of force relative to wave

sF Amplitude of in-phase force component

cF Amplitude of out-of-phase force component

sf , cf Normalized amplitudes of force components

M Wave induced moment

0M Amplitude of moment

M Phase angle of moment relative to wave

sM Amplitude of in-phase moment component

cM Amplitude of out-of-phase moment component

sm , cm Normalized amplitudes of moment components

M Inertia matrix

C Damping matrix

K Stiffness matrix

x State vector

y Output vector

u Periodic excitation vector

AΓ System state matrix

BΓ System input matrix

CΓ System output matrix

I Identity matrix

s Laplace transform variable

( )sG Transfer function matrix of heave and pitch dynamics

( )ZFG s Transfer function from force to heave

( )ZMG s Transfer function from moment to heave

( )FG s Transfer function from force to pitch

( )MG s Transfer function from moment to pitch

( )FG s Transfer function from wave to force

vii

( )MG s Transfer function from wave to moment

( )ZG s Transfer function from wave to heave

( )G s Transfer function from wave to pitch

( )Z eG j Magnitude of transfer function ( )ZG s at frequency e

( )Z eG j Phase of transfer function ( )ZG s at frequency e

( )eG j Magnitude of transfer function ( )G s at frequency e

( )eG j Phase of transfer function ( )G s at frequency e

1

1.0 Introduction The motivation for this research is to enhance a first principles, physics based, theoretical model to replicate small

planing boat motions as described in Cortesi, et. al. (2008) and Cortesi and Justh (2007). This model is based on

semi-empirical studies of planning hulls, such as those developed by Savitsky and Gore (1979) as well as the

principles discussed in Fossen (1994) regarding the modeling of displacement hulls. There has been some success

in using this relatively simple model; but it was deemed desirable to enhance the features of the model by adding

the effects of periodic wave action on the small boat. Some initial success was achieved in this endeavor by using

strip theory, as described in McCormick (1973) and Bhattacharyya (1978). However, the first-principles model only

worked for a limited range of boat speeds and wave frequencies. It was soon realized that for all of its simplicity,

the first model was unable to incorporate the complex effects of wave action. As a result, the NRL model was

supplemented for one which allows incorporation of the non-linear effects of a small boat undergoing motions due

to wave action as discussed in McCormick (1973), Bhattacharyya (1978), and McCormick (2010). Additionally, an

approach was needed which allows analytical data to be compared to experimental data over the full range of

encountering frequencies, e . It was therefore decided to use a frequency based approach. Using this approach,

the validity of the analytical model was able to be determined quickly and effectively.

This paper is organized into six sections. In Section 1.0, a brief introduction is given. Section 2.0 presents the

fundamentals of wave induced planar motions. Section 3.0 discusses the coupled equations of motion for heave

and pitch, and discusses the methods used to determine the coefficients for the equations of motion and the

derivation of the necessary transfer functions. It goes on to discuss the derivation of the vehicle parameters and

the derivation of the wave induced force and moment parameters. Section 4.0 discusses the methodology used to

conduct the validation experiment. Section 5.0 contains the results of the model and the experiment. In Section

6.0, a summary is given as well as recommendations for future work.

2.0 Review of Wave-Induced Planar Motion Analysis For this work, it is assumed that the vehicle is heading into the waves in a direction that is transverse to the wave

crest line. The seaway consists of regular harmonic waves, and the lateral vehicle motions are neglected. Heave

and pitch motions are referenced to the vehicle’s body axis that is rigidly located at the vehicle’s center of gravity,

CG. The planar geometry of the vehicle is illustrated in Figure (1). The heaving displacement, z, is the vertical

displacement of the CG from the still waterline; while pitch, θ, is the angular displacement of the vehicle’s x-y

plane from the still waterline. The free surface amplitude, ζ, is the vertical displacement of the water surface from

the still waterline measured at the vehicle’s CG.

________________Manuscript approved January 5, 2012.

2

Fig. 1 Heaving and Pitching Motion in Waves

The effective free-surface displacement encountered by the vehicle as it travels is described by

sin( )a et (1)

where a is the wave amplitude and e is the frequency of encounter. This frequency is related to the vehicle’s

velocity, V , the relative heading to the wave, , and the wave frequency, w ,according to the relationship

1 coswe w

V

g

(2)

For the case where the vehicle is heading into the wave train, the encountering angle is 180⁰.

The vehicle’s wave-induced heave and pitch motions are governed by the vehicle’s inertial interaction with the

hydrostatic and hydrodynamic forces acting on it. These interactions can be represented by two, coupled, second-

order differential equations with periodic forcing functions, as discussed in McCormick (1973), Bhattacharyya

(1978) and McCormick (2010),

( ) ( )zm a z bz cz d e h F t (3.a)

( ) ( )yy yyI A B C Dz Ez Hz M t (3.b)

The coefficients in Eqs. (3.a) and (3.b) correspond to the inertia, damping, and equivalent stiffness properties

associated with the vehicle-wave interaction. These coefficients are a function of the vehicle’s particular hull

shape, and are defined in Section 5.0. The terms ( )F t and ( )M t are the respective wave-induced excitation

force and moment. These terms will be discussed Section 6.0. Both the system coefficients and the excitation

terms are dependent on the encountered wave frequency, e . In addition, the excitation terms also depend on

the encountered wave amplitude, a . Thus, the heave and pitch dynamics are nonlinear in terms of frequency.

+z(t)

Aft

Still waterline

L

Fore

-θ(t)

+x

ζ

Wave direction Vehicle velocity

3

However, if we constrain our discussion to a particular encountering frequency, e , of interest the system

coefficients are constant and Eqs. (3.a) and (3.b) can be considered linear with regards to that frequency.

Therefore, Eq. (3) can be conveniently cast in compact matrix form, as in Goldstein (2001), Meirovitch (2010) and

Benaroya (2004).

( ) ( ) ( ) ( , )e e e e

z z zt

M C K u (4)

where the matrices are defined as follows:

The inertia matrix is

( )( )

( )

z

e

yy yy

m a d

D I A

M

while the damping matrix is

( )e

b e

E B

C

and the stiffness matrix is

( )e

c h

H C

K .

Also, in Eq. (4) is the periodic excitation vector,

( , )

( , )( , )

e

e

e

F tt

M t

u .

To analyze the relationship between the wave induced forces and the heaving and pitching motion, the two

second-order differential equations in Eq. (4) are formularized into a fourth-order, state variable model:

A Bx Γ x Γ u (5.a)

and

Cy Γ x (5.b)

where T

z z x is the state vector representing the entire state of the system at any given time, and

T

z y is output vector denoting the variables of interest, i.e., heave and pitch. The frequency dependent

systems parameters are contained in the system matrices defined below.

4

1 1

A

0 IΓ

M K M C,

1

B

0Γ

M , and CΓ I 0

(5.c)

3.0 Systems Analysis: Frequency Domain Model Knowing that the heaving and pitching dynamics are driven by the periodic wave function in Eq. (2), it is highly

advantageous to look at the wave-induced heave and pitch motions in the frequency domain. For a linear system,

such as the one represented by Eq. (5), a sinusoidal input of a specific frequency results in an output that is also a

sinusoid with the same frequency, but with a different amplitude and phase. The frequency-response function

describes the amplitude change and phase shift as a function of frequency. Thus, for a given seaway characterized

by Eq. (2) with a wave amplitude, a , and a frequency of encounter, e , the resulting heaving and pitching

motions will also be periodic with e and characterized by amplitudes az , a and phase angles z , as shown in

Figure (2).

Fig. 2 Input/Output Relationship between Wave and Vehicle Motion

To obtain a frequency-response function, employ the transfer function concept which relates the system output to

the system input in the Laplace domain. The transfer function concept allows one to convolve input/output

relationships algebraically, as in Meirovitch (1985). In the Laplace domain, Eq. (5) can be written in a compact

input-output relation as

( ) ( ) ( )s s sy G u (6)

where ( )sy and ( )su are the Laplace transforms of the output and input ( )ty and ( )tu , respectively. The term

( )sG is a transfer function matrix, which is an algebraic expression defined by the ratio

( )( )

( )

ss

s

yG

u (7)

The transfer function matrix ( )sG can be obtained from Eq. (5) by

1

( )

( ) ( )

( ) ( )

ZF ZM

F M

s s

G s G s

G s G s

C A BG Γ I Γ Γ

(8)

Resulting vehicle motion Encountered wave motion

Longitudinal

Dynamics

5

The elements of the matrix ( )sG are scalar transfer functions representing the interactions between heave (z),

pitch ( ), the wave induced force (F) and moment (M) as defined by ( )

( )( )

ZF

z sG s

F s ,

( )( )

( )ZM

z sG s

M s ,

( )( )

( )F

sG s

F s

,

( )( )

( )M

sG s

M s

, respectively. Therefore, the heave and pitch responses represented by Eq.

(6) can be written more explicitly as

( ) ( ) ( ) ( ) ( )ZF ZMz s G s F s G s M s (9)

( ) ( ) ( ) ( ) ( )F Ms G s F s G s M s (10)

The relationship between the respective wave-induced force and moment and the encountered waves can also be

expressed in the Laplace domain as

( ) ( ) ( )FF s G s s (11)

( ) ( ) ( )MM s G s s (12)

where transfer functions ( )

( )( )

F

F sG s

s

and

( )( )

( )M

M sG s

s

represent the dynamics of the wave force and

moment induced at the encountered waves. The derivation of the force and moment transfer functions, ( )FG s

and ( )MG s , is discussed in Section 6.0.

By substituting Eqs. (11) and (12) into Eqs. (9) and (10), the heaving and pitching motions are now related to the

encountered waves by

( ) ( ) ( ) ( ) ( ) ( )ZF F ZM Mz s G s G s G s G s s (13)

and

( ) ( ) ( ) ( ) ( ) ( )F F M Ms G s G s G s G s s (14)

respectively.

Expressing these relations in terms of transfer functions, one obtains

( )( )

( )

( ) ( ) ( ) ( )

Z

ZF F ZM M

z sG s

s

G s G s G s G s

(15)

and

6

( )( )

( )

( ) ( ) ( ) ( )F F M M

sG s

s

G s G s G s G s

(16)

For a sinusoidal wave input, where sin( )a et , it can be shown that when steady state conditions are

reached, the heaving and pitching frequency responses,

( ) sin( )ss a e zz t z t

( ) sin( )ss a et t ,

can be calculated by replacing the Laplace variable s in the transfer function Eqs. (15) and (16) by ej such that

( )a Z e az G j , ( )z Z eG j (17)

( )a e aG j , ( )eG j (18)

Here, 1j .

On this basis, ( )Z eG j and ( )eG j are amplitude ratios of heave and pitch resulting from the

encountered wave. Similarly, ( )Z eG j and ( )eG j are the heave and pitch phase-shift with respect to

the encountered wave. The amplitude ratios and phase shifts are functions of the encountered wave frequency

and therefore, can be conveniently presented in graphical form as illustrated later in the paper. These figures

show the heave amplitude ratio versus frequency of encounter. Given the wave amplitude and frequency for a

specific seaway, the resulting heave amplitude can be read directly from such a graph.

3.1 Vehicle Parameters The inertia, damping, and stiffness coefficients in Eq. (3) are defined in terms of strip theory as presented by

Krovin-Kroukovsky and Jacobs (1957). With this method, the vehicle is divided into a finite number of transverse

sections along its length. The inertia, damping, and stiffness coefficients for each section are considered constant

and are denoted by the subscript n. The overall vehicle coefficients are determined by integrating the section

coefficients over the vehicle length, L. The definitions of Eq. (3) coefficients are summarized in Table 1.

Table 1 Hydrodynamic Coefficient Definitions

Term Definition

n

L

m m d Vehicle mass

z n

L

a a d Added mass (the mass of water excited by the vehicle’s motion),

7

n

L

b b d Heave damping coefficient

n

L

c c d Heave stiffness coefficient

n

L

d a d Pitch inertia coupling term

n z

L

e b d ua Pitch damping coupling term

n

L

h c d ub

Pitch stiffness coupling term

2

yy n

L

I m d Pitch axis moment of inertia

2

yy n

L

A a d Added-mass of inertia

2

n

L

B b d Pitch damping coefficient

2

n

L

C c d uE Pitch stiffness coefficient

D d Heave inertia coupling term

n z

L

E b d ua Heave damping coupling term

n

L

H c d Heave Stiffness coupling term

The term nm in Table 1 is sectional vehicle mass. The term na is the sectional added mass and can be

determined from

21

8n a na k B (19)

where is the mass density, nB is the vehicle’s sectional beam width, and ak is the added mass coefficient. ak

is a function draft/beam ratio, shape of the hull’s cross sectional area, and the wave frequency, . Values of ak for

various hall cross section shapes can be found in Lewis (1929). The term nb is the sectional damping coefficient

and can be determined from

2 2

3

zn

e

g Rb

(20)

where g is the gravitational constant and zR is the ratio of radiated wave amplitude to heave motion amplitude.

The expression for zR used in this work was derived in Yamamoto et. al. (1980) and is given by

8

2

2

2 sin

ref ez

g n ez

BR e

g

(21)

The term refz is a sectional reference draft of the wetted surface.

The sectional stiffness coefficient, nc , is associated to the buoyancy restoring force due to heave disturbances and

is

n nc gB (22)

3.2 Wave Induced Force and Moment In general, the force induced by harmonic waves can be written as

0

( ) cos( ) sin( )

sin( )

c e s e

e F

F t F t F t

F t

(23)

where 0F is the force amplitude, defined by

2 2

0 c sF F F (24)

and F is the phase shift of the force with respect to the wave. That is,

1tan cF

s

F

F

(25)

The force amplitude terms cF and sF are derived in Goldstein (2001), Meirovitch (2010) and Benaroya (2004)

and can be expressed as

( ) ( )Kz

c e a c eF e f (26)

2( ) ( )sin( ) ( )cos( )n

c e n e n e n

L L

daf c a K d b u K d

d

(27)

( ) ( )Kz

s e a s eF e f (28)

2( ) ( )cos( ) ( )sin( )n

s e n e n e n

L L

daf c a K d b u K d

d

(29)

Substituting Eqs. (26) and (28) into Eq. (24) and dividing the resulting equation by the wave amplitude, one obtains

an expression for the amplitude ratios of force to wave. That is,

9

2 20 ( ) Kz

e c s

a

Fe f f

(30)

Similarly, substituting Eqs. (26) and (28) into Eq. (25) one obtains an expression for the phase shaft between the

force and wave which is,

1( ) tan cF e

s

f

f

(31)

The frequency transfer function between the wave induced force and the encountered waves can now be written

as

0( ) Fj

F e

a

FG j e

(32)

In a similar fashion, the moment induced by harmonic waves can be written as

0

( ) cos( ) sin( )

sin( )

c e s e

e M

M t M t M t

M t

(33)

where 0M is the moment amplitude defined by

2 2

0 c sM M M (34)

and M is the phase shift of the moment with respect to the wave, and is expressed as

1tan cM

s

M

M

(35)

The moment amplitude terms cM and sM can be expressed as

( ) ( )Kz

c e a c eM e m (36)

2( ) ( ) sin( ) ( ) cos( )n

c e n e n e n

L L

dam c a K d b u K d

d

(37)

( ) ( )Kz

s e a s eM e m (38)

2( ) ( ) cos( ) ( ) sin( )n

s e n e n e n

L L

dam c a K d b u K d

d

(39)

10

Substituting Eqs. (36) and (38) into Eq. (34) and dividing the resulting by the wave amplitude, an expression for the

amplitude ratios of moment to wave is obtained. That expression is

2 20 ( ) Kz

e c s

a

Me m m

(40)

Similarly, substituting Eqs. (36) and (38) into Eq. (35) an expression for the phase shaft between the moment and

wave is found to be.

1( ) tan cm e

s

m

m

(41)

The frequency transfer function between the wave induced moment and the encountered waves can now be

written as follows:

0( ) Mj

M e

a

MG j e

(42)

Eqs. (32) and (42) can now used with Eqs. (16) and (17) to develop the frequency response model.

4.0 Experiment To validate the model developed above, the wave-structure interaction of a small, shallow-draft, flat-bottom boat

was investigated in the 380 foot towing tank of the U. S. Naval Academy’s Hydromechanics Laboratory, described

in http://usna.edu/Hydromechanics/homepage.html. The test boat had the following characteristics:

Length = 3.05m

Beam = 1.142 m at L/2

Mass = 49.9 kg

To simulate the effects of onboard personnel and ancillary equipment, 120 kg of weights were added in the aft

section of the boat as sketched in Figure 3. Additionally, the boat was attached at its center to the towing carriage

via a set of heave posts which rode freely when any waves were encountered. The overall mass of the heave post

was 17.4 kg, making the overall mass of the boat 187.3 kg. Figure 4 shows a schematic of the Naval Academy

Towing Tank.

11

Fig. 3 Schematic of Test Boat

Fig. 4 Schematic Showing Plan and Elevation Views of Naval Academy Tow Tank

The 380-foot towing tank is large enough to test ship models up to 25 feet in length and weighing several thousand

pounds. The tank length allows testing at high speeds. The tank is outfitted with two towing carriages, a dual-flap

wavemaker and specialized equipment for measuring resistance, seakeeping and maneuvering characteristics of all

types of marine vehicles and ocean platforms. The test boat was attached to the towing carriage as seen below in

Figures 5 and 6.

56” 36”

120”

11”

120 kg

187.3 kg

Max Beam at Gunnels = 44.8”

Force Gage

2.5”

Tank Dimensions:

Length, Width, Depth

380 ft x 26 ft x16 ft

Towing Carriages

Maximum Speed: 32 ft/sec (9.8 m/sec)

Maximum Wave Height: 1.5 ft (0.46 m)

Wavemaker Wave Absorber - Beach

Plan View

Elevation View

12

Fig. 5 Photograph of the Test Boat Connected to Towing Carriage in Towing Tank

Fig. 6 View of the Test Boat from the Carriage

Twenty experimental runs were made, while varying three parameters: wave height, wave frequency, and ship

speed. A summary of the experimental runs is given in Table 2.

13

Table 2 Experimental Heave-to-Wave Amplitude Ratios

Boat Velocity

(Kts)

Wave Frequency

(Hz)

Encountered Wave Frequency

(Hz) Measured Heave-to-Wave Amplitude Ratio, a

a

z

1” Wave Height 2” Wave Height 3” Wave Height

3 0.4 0.56 0.992 1.088 1.105

3 0.7 1.18 0.752 0.712 -

3 1.0 1.99 0.140 0.109 -

4 0.4 0.61 1.035 1.085 1.141

4 0.7 1.35 0.510 0.723 -

4 1.0 2.32 0.152 0.180 -

6 0.4 0.72 1.196 1.361 1.397

6 0.7 1.67 0.885 0.860 -

6 1.0 2.98 - 0.275 -

5.0 Results In this section, experimental results are used to validate the dynamic model developed herein. The boat heave

and wave height for each experimental run was measured and recorded in time. To illustrate the experimental

results, the heave time responses at boat speeds of 3, 4, and 6 Kts to 1 inch, 0.4 Hz harmonic waves are shown in

Figures 7 through 9.

Fig. 7 Heave Time Response to 1 Inch - 0.4 Hz Waves, Vehicle Velocity = 3 Kts

4 6 8 10 12 14 16 18-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Time (sec)

He

av

e A

mp

litu

de

(m

)

Simulation

Experiment

14

Fig. 8 Heave Time Response to 1 Inch - 0.4 Hz Waves, Vehicle Velocity = 4 Kts

Fig. 9 Heave Time Response to 1 Inch - 0.4 Hz Waves, Vehicle Velocity = 6 Kts

4 6 8 10 12 14 16 18 20-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Time (sec)

He

av

e A

mp

litu

de

(m

)

Simulation

Experiment

2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Time (sec)

He

av

e A

mp

litu

de

(m

)

Simulation

Experiment

15

The experimental results show that the heave response to regular harmonic waves is also harmonic with frequency

equal to the encountered wave as expected. Also shown in Figures 7 through 9 are results of heave time response

simulations obtained from the heave dynamic model in Eq. (13). The measured time history of the wave height

was used as input for the simulations. As can be seen, the time simulation and the experimental results compare

well.

Knowing that the boat’s response to regular harmonic waves is also harmonic, it is highly advantageous to

represent the boat’s behavior as a frequency response. The frequency response conveniently characterizes the

boat’s response to a wider range of seaway conditions, as opposed to time responses. From the experimental

data, the ratio of the boat heave amplitude, az , and the wave amplitude, a , were calculated and shown in Table

2.

The experimental heave-to-wave amplitude ratios are plotted as a function of the frequency of encounter )( e

for the three boat velocities in Figures 10 to 12. The figures show that the heave amplitude ratio is fairly

independent to wave height changes at a given frequency. This result supports the linearity assumption of the

dynamics described in Eq. (3) at a specified input frequency.

Fig. 10 Heave Frequency Response to Wave Input, Vehicle Velocity = 3 Kts

10-1

100

101

-50

-40

-30

-20

-10

0

10

20

Am

p R

ati

o (

dB

)

Encountered Freq (Hz)

1" Wave Height

2" Wave Height

3" Wave Height

Experimental Data

16

Fig. 11 Heave Frequency Response to Wave Input, Vehicle Velocity = 4 Kts

Fig. 12 Heave Frequency Response to Wave Input, Vehicle Velocity = 6 Kts

10-1

100

101

-50

-40

-30

-20

-10

0

10

20

Am

p R

ati

o (

dB

)

Encountered Freq (Hz)

1" Wave Height

2" Wave Height

3" Wave Height

Experimental Data

10-1

100

101

-40

-30

-20

-10

0

10

20

30

Am

p R

ati

o (

dB

)

Encountered Freq (Hz)

1" Wave Height

2" Wave Height

3" Wave Height

Experimental Data

17

The frequency response amplitude ratio ( )Z eG j in Eq. (17) was calculated for the test boat, and is shown

superimposed on the experimental data in Figures 10 through 12. The frequency domain model shows the

characteristics associated with a fourth-order linear physical system. That is, constant low frequency amplitude

with resonate peaks around the natural frequency, and a -40 dB per decade slope at high frequencies. The heave

natural frequency can be calculated by

2n n

z

cf

m a

(43)

For the test boat, the natural frequency is approximately equal to fn = 0.64 Hz. The peaks and valleys observed at

frequencies greater than n correspond to nonlinearities in the buoyancy force and moment. Buoyancy depends

heavily on the relationship between wave length and boat length. Buoyancy will decrease when the boat length is

an integer multiple of the wave length and will increase when it is not. Therefore, as the wave frequency increases

and the wave length decreases, the buoyancy force and moment will rise and fall.

In general, the frequency domain model corresponds well with the experimental data, especially at low

frequencies. Any discrepancies observed between the model and data can be attributed to inaccuracies in the

model’s added mass and damping coefficient terms. Expressions for these terms were derived from interpolations

of experimental data and therefore, inherently have a degree of uncertainty. In addition, experimental error

associated with the towing tank experiments may have also contributed to any discrepancies.

6.0 Conclusions In this paper, a model for the coupled heave and pitch dynamics of surface vehicles is developed using a novel

approach of representing the ship dynamics in the frequency domain. The model is based on theoretical first

principles involving highly nonlinear equations that depend on the wave frequency. Compared to time domain

modeling and simulation approaches, the frequency domain model provides greater insight to the surface vehicle’s

behavior over a wider operating range. The theoretical model was validated by experimental data that was

obtained from towing tank experiment. The test boat has a 3-meter shallow draft and a flat bottom. The

theoretical model results compare well with the experimental data.

Acknowledgements The authors would like to thank Dr. Michael McCormick for his outstanding technical advice and encouragement

during this effort. The authors also thank the dedicated personnel in the United States Naval Academy’s

Hydromechanics Laboratory for their invaluable assistance with the experiment associated with this paper.

18

References Bhattacharyya, R. - (1978) Dynamics of Marine Vehicles, Wiley-Interscience, New York.

Benaroya, H. (2004) - Mechanical Vibration – Analysis, Uncertainties, and Control, Marcel Dekker, New York.

Cortesi, R.S., Galloway, K.S., and Justh, E.W. (2008) - "A biologically inspired approach to modeling unmanned

vehicle teams," Evolutionary and Bio-inspired Computation: Theory and Applications II, M. Blowers and A.F.

Sisti, Eds., Proc. SPIE-The International Society for Optical Engineering, Vol. 6964, pp. 696405-1 to 696405-

12,.

Cortesi, R.S. and Justh, E.W. (2007) “The NRL small planing boat model,” Naval Research Lab preprint,.

Savitsky, D. and Gore, J.L. (1979) - “Re-evaluation of the planning hull form,” Journal of Hydronautics 14(2), pp. 34-

47.

Fossen, T.I. (1994) - Guidance and Control of Ocean Vehicles, John Wiley and Sons.

Goldstein, H. (2001) - Classical Mechanics, Addison Wesley, New York.

Krovin-Kroukovsky, B.V. and Jacobs, W. R. (1957) - “Pitching and Heaving Motions of a Ship in Regular Waves”,

Transactions of Society of Naval Architects and Marine Engineers, Vol. 65, pp. 590-632.

Lewis., F. M. (1929) - “The Inertia of the Water Surrounding a Vibrating Ship”, Transactions of Society of Naval

Architects and Marine Engineers.

McCormick, M. E. (1973) - Ocean Engineering Wave Mechanics, Wiley-Interscience, New York.

McCormick, M. E. (2010) - Ocean Engineering Mechanics with Applications, Cambridge University Press, New York.

Meirovitch, L. (2010) - Fundamentals of Vibrations, Waveland Press, Illinois.

Meirovitch, L. (1985) - Introduction to Dynamics and Control, Wiley, New York.

United States Naval Academy Hydromechanics Laboratory Web Page,

http://usna.edu/Hydromechanics/homepage.html,

Yamamoto, Y., Fujino, M., and Fukasawa, T. (1980) - “Motion and Longitudinal Strength of a Ship in Head Sea and

the Effects of Non-Linearities” Naval Architecture and Ocean Engineering, Society of Naval Architects of

Japan, Vol. 18.